NXT is powered by 6 alkaline cells that drain
rather fast for heavy duty robots. Your pocket money may follow
the same curve if you use your NXT for extended periods! You
have a few options to avoid that:

Use NiMH rechargeable batteries instead
of alkaline batteries. This is a good solution for mobile
robots but it has some shortcomings. You need to keep a
pack of batteries charged when the first is empty, charging
6 cells is not easy since most chargers have only 4 slots,
and the 7.2V voltage obtained with 6 NiMH batteries seriously
reduces the power and rotation speed of NXT motors (see
performance charts).

Use LEGO
Li-ion battery pack. A very interesting product, with
some shortcomings too... The battery pack is bigger than
the batteries cover it replaces, meaning that some robots
that have parts just behind the regular battery lid are
impossible to build.

Build your own mains adapter, as described
here. No autonomous robots with this one, but full power
at economical price for your static constructions...

WARNING: Do not try to build this adapter
if you don't know what you are doing.I shall not be held responsible if you burn your NXT
with incorrect voltage or bad polarity !

Schematics

The schematics is the easy part...
A power jack socket is connected to NXT supply contacts
through optional protection elements. F1 is a 3A
fast blow fuse (rating should match intensity that
can deliver your mains adapter). D1 is a transient
suppressor diode, 9V rated. It should protect
your NXT from mains adapter failure (or from plugging
the wrong plug in your NXT!). The one I used is
a Fairchild SMBJ9V0A.

These protection components are not really necessary,
but I strongly suggest that you mount them to avoid
problems.

Mains
adapter selection

I selected a switch mode power supply,
that delivers a clean, well regulated 9V tension.

It can deliver a hefty 3.6A, more
than enough for the NXT. I suggest to choose an
adapter that can deliver at least 2A to be able
to supply heavy duty models.

Note that this adapter has a switchable voltage
output, make sure you don't exceed 9V. The motor
drivers which are connected directly to that supply
are specified for 9V supply and should never be
powered at more than 10.5V.

Similarly the plug of these supplies
often have a polarity selection option, and documentation
for that is generally terse, or even misleading.
I suggest that you verify with a voltmeter that
the polarity is conform to your expectation! The
most common choice is th have the + in center, -
on plug tube, that's the one I use for this project.

Warning:
Cheap transformer based wall adapters,
such as the one photographed here, are NOT suited
for this use: their regulation is poor at best,
and voltage at light load is much higher than 9V.
Moreover, at the power level required here, they
are bulky and very heavy.

Mechanical
construction

The mechanical construction requires
some care to get a supply plate that fits well NXT
bottom. We start from a blank FR4 (glass loaded
epoxy) single sided PCB plate, the small piece will
be glued to form plate hinge. Two areas are electrically
separated with a knife from the main one (take care
of your fingers!) to support fuse and diode.

The board dimensions. The two notches
at bottom, not shown on photo above, are made with
a metal saw and will receive the retainer clip.

The socket and the hinge are epoxy-glued
to the board. Since my hinge was slightly too large,
I had to trim it after mounting (hole in center).
The hole in the socket must be centered.

Metallic battery holder clips are
used to connect NXT batteries contacts. I used Keystone
596 clips. The bottom retainers are cut before mounting.
The clip with spring was not available at the time...

...so I scavenged a spring from a
cheap battery holder and soldered it to a 596 clip.

Positions of socket and clips.

All the elements are surface-soldered
to the board. As the copper area of the board is
large, it makes a good heatsink. Use a powerful
iron with a large tip and high temperature.

Detail of the diode area.

The plug and matching socket terminals.

The
paper clip solution

I wondered for a while
how to emulate the latch that hold battery lid...
a bent paper clip provided the solution

It is inserted in the
two notches of the board (so it doesn't protrude
on either side of the plate)...

...and soldered (and/or
glued) there.

Clip dimensions.

Completed
replacement plate

Ready for insertion.

Lid closed.

The retaining clip.
Precise shape has to be trimmed with pliers...

Just plug... and play!

WARNING: Do not try to build this adapter
if you don't know what you are doing.I shall not be held responsible if you burn your NXT
with incorrect voltage or bad polarity !